Abstract

In this paper a solution of the direct Cauchy problems for heat equation is founded in the form of Hermite polynomial series. A well-known classical solution of direct Cauchy problem is represented as Poisson's integral. The author reveals, the formulas obtained by him for solution of the inverse Cauchy problems have a symmetry with respect to the formulas for corresponding direct Cauchy problems. Obtained formulas for solution of the inverse problems can serve as a basis for reg-ularizing computational algorithms while well-known classical formula for the solution of the inverse Cauchy problem can't be a basis for regu-larizing computational algorithms.

1. Introduction

In this paper a direct and an inverse Cauchy problems for the heat equation are solved at Cartesian and polar coordinates. The inverse Cauchy problem for the heat equation consists of to reconstruct a priori unknown initial condition of the dynamic system from its known final condition. In 1939 French mathematician Jacques Hadamard defined the problem is called well-posed if a solution exists, the solution is unique, the solution’s behavior hardly changes when there’s a slight change in the initial condition. The problems are called ill-posed or not well-posed if at least one of these three conditions is not fulfilled. The most often, the third condition so called the stability condition of solution is violated for ill-posed problems. In this case there is a paradoxical situation: the problem is mathematically defined but solution cannot be obtained by conventional methods. In mathematics the vast majority of inverse problems is not well-posed: small perturbations of the initial data (observations) may correspond to an arbitrarily large perturbations of the solution. A classic example of ill-posed problem is the inverse Cauchy problem (retrospective problem) for the heat equation. The direct Cauchy problem for the heat equation is well-posed most frequently. In this work the solution for direct Cauchy problems found in the form of the Hermite polynomial series. A wellknown classical solution for the direct Cauchy problem is represented in the form of Poisson’s integral. We noticed the solution of inverse Cauchy problem possess the symmetry with respect to solution of the direct Cauchy problem. Formulas for the inverse problem obtained in this paper can serve as a basis for regularizing computational algorithm. Previously known classical formula for the solution of the inverse problem did not have such properties, that’s why classical formula can’t serve as a basis for convergent algorithm. The main result for the direct problems is formulas (5), (6), (9), (23), (24), (25). The main result for the inverse problems is formulas (13), (14), (15), (27), (28), (29).

2. The Cauchy Problem (Direct Problem) for the Heat Equation

The solution of the Cauchy problem with the initial thermal field f (x) for an infinite bar we will get as the Hermite polynomial series. In order to get this result, we use the well-known analytic solution in the form of Fourier’s integral, [1]:

where f (x) - the initial thermal field, - thermal field in time and at the point x.

Write the last equality in the form

(1)

where β > 0.

The function is a generating function for the Hermite polynomials, [1], this means that

(2)

where

(3)

the Hermite polynomials.

In accordance with (2) the formula (1) takes the form

If we may change the order of integration, then compute inner integral with respect to λ. We use the Poisson’s integral, [13]. We get

(4)

Differentiating both sides of ( 4 ) with respect to x we obtain the value of the desired integral

On the basis of Hermite polynomials (3) the last formula can be written

Finally, we get an analytical representation for the thermal field in time and at the point x

(5)

where

Remark. By interchanging the variables x and ξ in (5) we get another version of the thermal field:

(6)

where

Now we get the third new formula. To do this, the formula (1) can be written in the form

where β > 0.

In view of (2) we get

To simplify last formula, we change the order of integration and compute inner integral with respect to λ, substitute x = 0 in (4). We obtain

The inverse problem for the heat equation of an infinite bar is to find the unknown initial distribution f (x) of thermal field by the known temperature field [3, 4, 5, 7, 8]. This problem leads to the solving of first - type Fredholm integral equation:

(10)

The left side of equation (10) is Poisson’s integral, [15]. As it is shown in [1, 2] the solution of equation (10) is:

(11)

where Hj(z) - Hermite polynomials, (3).

Formula (11) contains a derivatives of an arbitrarily high order so formula (11) can’t serve as a basis for the regularizing computational algorithm. Consequently, it is actual to find the new formulas without derivatives for the solving of the equation (10).

As in section-1, we obtain three new formulas. We get the solution of equation (10) by the Fourier transform integral method from [1, 9, 11, 12]

If β > 0, then the last formula takes the form

(12)

Because of formula (2) we get

We will change the order of integration and calculate inner integral with respect to λ. The Poisson integral (4) [15] is used in this calculations. We get

We calculate the value of an integral by j - time differentiating with respect to x:

On the basis of (3) we can write

Finally, first new formula for the initial distribution of the thermal field takes the form

(13)

where

Remark. By interchanging variables x and ξ in (13) we get a second new formula for the solution of equation (10):

(14)

where

Finally, we will prove the third new formula for the inverse Cauchy problem. We use (12) which can be written as

where β > 0. Because of formula (13) we get

If we use formula (4), then the initial distribution of the thermal field takes the form

where

Because of formula (8) for Hj(0) as a result we get

(15)

Where

3. Cauchy Problem for the Heat Equation at Polar Coordinates

3.1. Auxiliary Statements

We define polynomials Wj(z) with the help of the generating function

(16)

where I0(x)- the zero-order modified Bessel function of the first kind. It follows from (16) that the polynomials Wj(z) have the form

(17)

We get another view of polynomials Wj(z). From the definition of the Bessel operator, [10],

it follows that

therefore

In this formula we equate the coefficients of ξ2j- degree at the left and right sides. We get

hence the polynomials Wj (z) have the form

Using polynomials Wj (z) we will obtain the new formulas for solutions of direct and inverse Cauchy problems at polar coordinates.

3.2. New Formulas for Solution of the Cauchy Problem at Polar Coordi-Nates

We deduce the new formulas for the solution of the Cauchy problem for theheat equation at polar coordinates (r, ϕ) if the thermal regime depends on only thevariable r. We use the explicit formula for solution of the Cauchy problem at polarcoordinates [6]

where J0(λξ) - the zero-order Bessel function of the first kind [1]. We write the last equation in the form

(18)

where β > 0. In (16) we make the substitution: , Then we obtain

(19)

Formula (18) according to the relation (19) takes the form

(20)

In last formula we change the order of integration and compute the inner integral with respect to λ. The Weber integral [6] is used in calculation.

Equating the coefficients at ξ2j-degree we get the value of an integral

(21)

Due to (17) the last formula can be written:

Then, (21) becomes

(22)

Finally, formula (20) for the thermal field with (22) takes the form

(23)

where

Remark. From the last equality for at the expansion theorem with respect to eigenfunctions can be obtained

where

Similarly section-1 we can get two new formulas for the thermal field. If we replace in (23): , then the new formula takes the form

(24)

where

The third new formula is proved similarly section-1. We apply formula from [10]

to (18). Then we get

(25)

Write equation (18) as

where β > 0. Next, on the basis of (25) we have

We change the order of integration and use the definition of the Gamma-function [10] to compute the inner integral with respect to λ.

Finally, we obtain

(26)

where

4. The Inverse Cauchy Problem for the Heat Equation at Polar Coordinates